Abstract

Multiconjugate adaptive optics (MCAO) systems with 104–105 degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wavefront control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction matrix scales respectively with the cube and the square of the number of adaptive optics degrees of freedom. We develop scalable open-loop iterative sparse matrix implementations of minimum variance wave-front reconstruction for telescope diameters up to 32 m with more than 104 actuators. The basic approach is the preconditioned conjugate gradient method with an efficient preconditioner, whose block structure is defined by the atmospheric turbulent layers very much like the layer-oriented MCAO algorithms of current interest. Two cost-effective preconditioners are investigated: a multigrid solver and a simpler block symmetric Gauss-Seidel (BSGS) sweep. Both options require off-line sparse Cholesky factorizations of the diagonal blocks of the matrix system. The cost to precompute these factors scales approximately as the three-halves power of the number of estimated phase grid points per atmospheric layer, and their average update rate is typically of the order of 10-2 Hz, i.e., 4–5 orders of magnitude lower than the typical 103 Hz temporal sampling rate. All other computations scale almost linearly with the total number of estimated phase grid points. We present numerical simulation results to illustrate algorithm convergence. Convergence rates of both preconditioners are similar, regardless of measurement noise level, indicating that the layer-oriented BSGS sweep is as effective as the more elaborated multiresolution preconditioner.

J. M. Beckers, “Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,” in Proceedings of the European Southern Observatory Conference and Workshop on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed., 30, 693–703 (1998).

Barker, V. A.

Beckers, J. M.

J. M. Beckers, “Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,” in Proceedings of the European Southern Observatory Conference and Workshop on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed., 30, 693–703 (1998).

J. M. Beckers, “Increasing the size of the isoplanatic patch with multiconjugate adaptive optics,” in Proceedings of the European Southern Observatory Conference and Workshop on Very Large Telescopes and their Instrumentation, M. H. Ulrich, ed., 30, 693–703 (1998).

Square matrix to be inverted for the fitting step (sparse component), for a system with Ndm = 3 DMs. Dashed horizontal and vertical lines have been drawn to display the block structure of the matrix. Each block’s size is Na/Ndm × Na/Ndm = 3408 × 3408, corresponding to 3408 actuators per DM. Total matrix size is thus Na × Na = 10,224 × 10,224.

Fitting step for the same MCAO system as in Fig.
12. Ndm = 3 DMs at altitudes 0 m, 5150 m, and 10,300 m. All DMs are at half-atmospheric phase screen resolution. Plain CG applied to the sparse component of the matrix to be inverted for the fitting step. Regularization parameter α = 10-5. Wide-field rms residual error averaged over the Nobs = 25 observation directions given in Fig.
6 and five simulation runs.

Top panels: estimation step convergence comparison for the BSGS, MG preconditioners, and no preconditioning. The 32-m MCAO system operating at SNR = 100. Results were averaged over the Nobs = 25 observation directions given in Fig.
6 and five simulation runs. Bottom panels: plain CG for the fitting step.

Convergence comparison for the 32-m MCAO system operating at two different measurement noise levels given by SNR = 1 and SNR = 20. The estimation step used the BSGS preconditioner. Results were averaged over Nobs = 25 observation directions partitioning the FoV and five simulation runs.